An extension problem in equivariant cohomology
Let \(G\) be a group, \(BG\) its classifying space and \(X\) be a \(G\)-space. We denote by \(X_{hG} = EG \times_G X\) the homotopy quotient of \(X\) and by \(H^*_GX=H^*(X_{hG};\mathbb F_2)\) the mod-2 equivariant cohomology of \(X\). The module \(H^*_GX\) is an object of the category \(H^*BG-{\mathcal U}\) of unstable \(H^*BG-A\)-modules; \(A\) denotes the mod-2 Steenrod algebra and \(H^*_GX\) is an \(H^*BG\)-module via the morphism induced in mod-2 cohomology by the projection \(X_{hG} \rightarrow \{*\}_{hG}=BG\). Let \(\overline{H^*_GX}= \mathbb F_2\otimes_{H^*BG} H^*_GX\), \(\mathbb F_2\) is considered as a trivial \(H^*BG\)-module. In this paper the author studies the surjectivity of the homomorphism: \[ \Hom_{H*BG-{\mathcal U}}(H^*_GY, H^*_GX)\rightarrow \Hom_{\mathcal U} (\overline{H^*_GY}, \overline{H^*_GX}),\quad f \mapsto \overline{f} \] when the group \(G\) is an elementary abelian 2-group.
scientific article; zbMATH DE number 5547401